Exact analytical results are presented for the neutralization integral in LEIS. These results generalize the results of Godfrey-Woodruff and Richard-Eschenbacher by (i) considering a Coulomb trajectory as well as the usual asymptotic, i.e. free particle, trajectory, and (ii) introducing new forms for the neutralization rate. These forms might be relevant parameterizations for reionization as well as neutralization. The analytical results for several ion trajectories are compared to numerical results for the Thomas-Fermi-Molière trajectory. In conclusion, the implementation of these results in simulation codes is discussed.